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| In mathematics, '''Humbert series''' are a set of seven [[hypergeometric series]] Φ<sub>1</sub>, Φ<sub>2</sub>, Φ<sub>3</sub>, Ψ<sub>1</sub>, Ψ<sub>2</sub>, Ξ<sub>1</sub>, Ξ<sub>2</sub> of two [[variable (mathematics)|variable]]s that generalize [[confluent hypergeometric function|Kummer's confluent hypergeometric series]] <sub>1</sub>''F''<sub>1</sub> of one variable and the [[confluent hypergeometric limit function]] <sub>0</sub>''F''<sub>1</sub> of one variable. The first of these double series was introduced by {{harvs|txt|authorlink=Pierre Humbert (mathematician)|first=Pierre|last= Humbert|year=1920}}.
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| ==Definitions==
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| The Humbert series Φ<sub>1</sub> is defined for |''x''| < 1 by the double series:
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| :<math>
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| \Phi_1(a,b,c;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~,
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| </math>
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| where the [[Pochhammer symbol]] (''q'')<sub>''n''</sub> represents the rising factorial: | |
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| :<math>(q)_n = \frac{\Gamma(q+n)}{\Gamma(q)} = q\,(q+1) \cdots (q+n-1) ~.</math>
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| For other values of ''x'' the function Φ<sub>1</sub> can be defined by [[analytic continuation]].
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| Similarly, the function Φ<sub>2</sub> is defined for all ''x'', ''y'' by the series:
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| :<math>
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| \Phi_2(b_1,b_2,c;x,y) = \sum_{m,n=0}^\infty \frac{(b_1)_m (b_2)_n} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~,
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| </math>
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|
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| the function Φ<sub>3</sub> for all ''x'', ''y'' by the series:
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| :<math>
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| \Phi_3(b,c;x,y) = \sum_{m,n=0}^\infty \frac{(b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~,
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| </math>
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| the function Ψ<sub>1</sub> for |''x''| < 1 by the series:
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| :<math>
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| \Psi_1(a,b,c_1,c_2;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n} (b)_m} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n ~,
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| </math>
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| the function Ψ<sub>2</sub> for all ''x'', ''y'' by the series:
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| :<math>
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| \Psi_2(a,c_1,c_2;x,y) = \sum_{m,n=0}^\infty \frac{(a)_{m+n}} {(c_1)_m (c_2)_n \,m! \,n!} \,x^m y^n ~,
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| </math>
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| the function Ξ<sub>1</sub> for |''x''| < 1 by the series:
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| :<math>
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| \Xi_1(a_1,a_2,b,c;x,y) = \sum_{m,n=0}^\infty \frac{(a_1)_m (a_2)_n (b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~,
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| </math>
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| and the function Ξ<sub>2</sub> for |''x''| < 1 by the series: | |
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| :<math> | |
| \Xi_2(a,b,c;x,y) = \sum_{m,n=0}^\infty \frac{(a)_m (b)_m} {(c)_{m+n} \,m! \,n!} \,x^m y^n ~.
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| </math>
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| ==Related series==
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| * {{main|Appell series}}
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| :There are four related series of two variables, ''F''<sub>1</sub>, ''F''<sub>2</sub>, ''F''<sub>3</sub>, and ''F''<sub>4</sub>, which generalize [[hypergeometric function|Gauss's hypergeometric series]] <sub>2</sub>''F''<sub>1</sub> of one variable in a similar manner and which were introduced by [[Paul Émile Appell]] in 1880.
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| ==References==
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| * {{cite book | last1= Appell | first1= Paul | author1-link= Paul Émile Appell | last2= Kampé de Fériet | first2= Joseph | author2-link= Joseph Kampé de Fériet | title= Fonctions hypergéométriques et hypersphériques; Polynômes d'Hermite | language= French | location= Paris | publisher= Gauthier–Villars | year= 1926 | jfm= 52.0361.13 | ref= harv}} (see p. 126)
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| * {{cite book | first1= H. | last1= Bateman | author1-link= Harry Bateman | first2= A. | last2= Erdélyi | author2-link= Arthur Erdélyi | title= Higher Transcendental Functions, Vol. I | url= http://apps.nrbook.com/bateman/Vol1.pdf | format= PDF | location= New York | publisher= McGraw–Hill | year= 1953 | ref = harv}} (see p. 225)
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| * {{cite book | last1= Gradshteyn | first1= I. S. | last2= Ryzhik | first2= I. M. | title= Table of integrals, series, and products | edition= 7th | publisher= Elsevier/Academic Press, Amsterdam | year= 2007 | isbn= 978-0-12-373637-6 | mr= 2360010 | ref = harv}} (see Chapter 9.26)
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| * {{cite journal | last= Humbert | first= Pierre | authorlink= Pierre Humbert (mathematician) | title= Sur les fonctions hypercylindriques | language= French | journal= Comptes rendus hebdomadaires des séances de l'Académie des sciences | year= 1920 | volume= 171 | pages= 490–492 | jfm= 47.0348.01 | ref= harv}}
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| {{DEFAULTSORT:Humbert Series}}
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| [[Category:Hypergeometric functions]]
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| [[Category:Mathematical series]]
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